Download The 12th Delfino Problem and universally Baire sets of reals

The 12th Delfino Problem and universally Baire
sets of reals
Philipp Doebler
[email protected]
Abstract
We prove a theorem of Steel that solves the 12th Delfino Problem. We
build on the one hand on a lemma of Woodin on universally Baire sets and
their projections in certain generic extensions in the presence of strong
cardinals; on the other hand we use certain premice to find projective
uniformizations of projective sets.
1
Introduction
The following is a translation of the last two chapters of the author’s Diplomarbeit (Master’s Thesis). Since this Diplomarbeit was mostly concerned with
universally Baire sets of reals, the first part of the following deals with universally
Baire sets in the presence of strong cardinals. The second part discusses the 12th
Delfino Problem. For the reader’s convenience we have included some definitions
and some results. For details on universally Baire sets we recommend the original
paper by Feng, Magidor and Woodin [FMW92].
The author would like to thank Ralf Schindler for the excellent advice provided
during the work on the Diplomarbeit mentioned above. The Diplomarbeit is
based on handwritten notes by Ralf Schindler. None of the material in this paper
is due to the author.
We will give a definition of (λ-)universally Baire now. Note that this is not the
original definition given by Feng, Magidor and Woodin but rather an equivalent
one; the equivalence was proved in [FMW92]. Versions of all the results in the
introduction can also be found in [FMW92].
Definition 1.1 Let A ⊆ ω ω and λ an infinite cardinal. We will call A λuniversally Baire if and only if there are trees T and T ∗ such that for every notion
of forcing P of cardinality equal or less than λ the following three conditions hold:
1. A = p[T ], ω ω − A = p[T ∗ ],
2. P p[Ť ] ∪ p[Tˇ∗ ] = ω ω ,
1
3. P p[Ť ] ∩ p[Tˇ∗ ] = ∅.
We will call A universally Baire, if it is λ-universally Baire for all infinite cardinals λ. We remark that the first condition implies the third by absoluteness of
wellfoundedness; since the tree
S := {(s, f, g) ∈ ω <ω × (λ<ω )2 ; (s, f ) ∈ T ∧ (s, g) ∈ T ∗ }.
searching for a branch through T and T ∗ is wellfounded (in V ), it is wellfounded
in every other model containing T and T ∗ .
Note that the original definition easily implies that every ω-universally Baire
set has the property of Baire. The way we have formulated the definition makes
this a nontrivial fact.
We can verify that a given set of reals is universally Baire by verifying the definition for the adequate Levy collapse:
Theorem 1.2 (Feng, Magidor, Woodin) Let A ⊆ ω ω and λ and infinite cardinal.
The following are equivalent:
1. There are two trees T and T ∗ such that:
(a) A = p[T ], ω ω − A = p[T ∗ ],
(b) Col(ω, λ) p[Ť ] ∪ p[Tˇ∗ ] = ω ω ,
(c) Col(ω, λ) p[Ť ] ∩ p[Tˇ∗ ] = ∅.
2. A is λ-universally Baire.
Remark 1.3 The trees T and T ∗ in the preceding theorem can be chosen to be
trees on ω × 2λ .
What pointclasses are universally Baire? This question depends on large
cardinal assumptions. In ZF C alone we know the following:
Proposition 1.4 The analytic sets (and hence the coanalytic sets) are universally Baire. Without loss of generality the tree for a (lightface) Π11 set is
constructible, i.e. it is in L and is the appropriate Shoenfield tree.
This result is optimal in the following sense:
Proposition 1.5 Every ω-universally Baire set has the Baire property and every
2ω -universally Baire set is Lebesgue measurable. Thus in L there exists a ∆12 set
that is not universally Baire.
2
The presence of certain large cardinals allows us to construct trees witnessing
that every member of certain pointclasses of sets of reals is (λ-)universally Baire.
One example of a result of this type is the following theorem.
Theorem 1.6 (Feng, Magidor, Woodin) The following are equivalent:
1. Every Σ12 set is universally Baire.
2. For every set x x] exists.1
Another type of axiom that influences which pointclasses are universally Baire
are axioms of the type “V is Γ-absolute in relation to some class of set generic
extensions of V ” where Γ is a suitable subclass of the projective sets. We give an
example of this type of result.
Theorem 1.7 (Feng, Magidor, Woodin) Let λ ≥ ω be an infinite cardinal. The
following are equivalent:
1. If ϕ(x1 , . . . , xn ) is a Σ13 formula with free variables ranging over x1 , . . . , xn
and a1 , . . . , an ∈ ω ω then the following holds true:
ϕ(a1 , . . . , an ) ⇐⇒ Col(ω, λ) ϕ(aˇ1 , . . . , aˇn ).
2. Every ∆12 set is λ-universally Baire.
2
Strong cardinals an universally Baire sets
In this chapter we will discuss a tree construction in the presence of a strong
cardinal due to Woodin. This construction yields that projections of universally
Baire sets are again universally Baire in a certain generic extension. To be more
precise: Let A ⊂ (ω ω )2 be universally Baire. We will construct a generic extension
that contains trees witnessing that ∃R A := {x ; ∃y(x, y) ∈ A} is universally Baire
in this generic extension.
2.1
Strong cardinals
Definition 2.1 Let λ ≥ κ. A cardinal κ is λ-strong if and only if for all X ∈ Hλ+
there is a transitive M ⊂ V and an embedding π : V → M such that X ∈ M and
κ = crit(π) and π(κ) > λ. A cardinal κ is strong, if it is λ-strong for all λ.
1
Since we are just quoting this result to give an example of a large cardinal assumption that
influences which pointclasses are universally Baire, we will not go into the details about sharps.
3
Definition 2.2 Let M ⊂ V be transitive. Let π : V → M be a nontrivial
elementary embedding; we denote its critical point by crit(π) = κ. Let λ > κ.
The (κ, λ)-extender E derived from π is defined as follows:
• E = {Ea ; a is a finite partial function ω → λ},
• Ea ⊆ P(κdom(a) ),
• X ∈ Ea ⇐⇒ a ∈ π(X).
Every Ea is hence a κ-complete ultrafilter on P(κdom(a) ). Let b be a finite
partial function ω → λ and a ⊆ b. Let X ∈ P(κdom(a) ). We define
X ab = {x ∈ κdom(b) ; x dom(a) ∈ X}.
Then
X ∈ Ea ⇐⇒ X ab ∈ Eb .
The existence of an embedding π as in definition 2.1 can be traced back to the
existence of an adequate extender, hence to the existence of a set (see [Jec03,
20.30]). We could thus have avoided classes in the definition of (λ-)strong cardinals, but the above definition is more natural.
2.2
Strong cardinals and tree constructions
κ
Lemma 2.3 (Woodin) Let κ be λ-strong and 22 < λ. Let A ⊆ (ω ω )2 be λuniversally Baire. Let T and T ∗ denote the trees, that witness that A = p[T ]
κ
is λ-universally Baire. Let H be Col(ω, 22 )-generic over V . Then in V [H] the
following holds: There are trees U and U ∗ , that witness the λ-universal Baireness
of ∃R A = p[U ].
Proof. By theorem 1.2 it suffices to discuss the forcing Col(ω, λ). We set P =
Col(ω, λ). We need to show, that there are trees U and U ∗ in V [H], such that
for every G P-generic over V [H]
V [H][G] |= p[U ] = ω ω − p[U ∗ ].
By remark 1.3 the trees T and T ∗ are, without loss of generality, trees on ω×ω×2λ .
Clearly P ∈ Hλ+ . Let π : V → M be an embedding with critical point κ such,
κ
κ
that P(Col(ω, 22 )) ∈ M . Note that H is Col(ω, 22 )-generic over M as well. By
[Kan03, 13.13] there is a tree U such that p[U ] = ∃R p[T ]. The tree U can be
chosen to be a tree on ω × 2λ .
Claim 1. If K is P-generic over V [H], then
V [H][K] |= p[U ] = p[π(U )].
4
Proof of Claim 1. Let x ∈ V [H][K] ∩ ω ω . We discuss the case x ∈ p[U ], say
(x, y) ∈ p[T ]. Then there is f : ω → 2λ such that
∀n(x n, y n, f n) ∈ T.
Hence for arbitrary n ∈ ω
(x n, y n, π(f n)) ∈ π(T ),
and so x ∈ p[π(U )] as witnessed by π 00 {f n ; n ∈ ω} : ω → π(2λ ).
Now let x ∈
/ p[U ]. Hence ∀y(x, y) ∈
/ p[T ]. Since T and T ∗ are trees of a λuniversally Baire set and V [H][K] is a generic extension of size ≤ λ, we have
V [H][K] |= ∀y(x, y) ∈ p[T ∗ ].
An argument as in the above case yields ∀y(x, y) ∈ p[π(T ∗ )]. Since
V |= p[T ] ∩ p[T ∗ ] = ∅ in all generic extensions of size ≤ λ,
the elementarity of π implies
M |= p[π(T )] ∩ p[π(T ∗ )] = ∅ in all generic extensions of size ≤ π(λ).
We reformulate this fact: The tree
T̃ := {(s, t, f, g) ∈ ω <ω ×ω <ω ×π(λ)<ω ×π(λ)<ω ; (s, t, f ) ∈ π(T )∧(s, t, g) ∈ π(T ∗ )}
is well-founded. T̃ is the tree that searches for a“common branch” through both
π(T ) and π(T ∗ ). Wellfoundedness is absolute, so
V [H][K] |= p[π(T )] ∩ p[π(T ∗ )] = ∅.
In particular ∀y(x, y) ∈
/ p[π(T )], hence x ∈
/ p[π(U )].
(Claim 1)
We now have to look for a tree U ∗ in V [H] such that for every K that is
P-generic over M [H]
M [H][K] |= p[π(U )] = ω ω − p[U ∗ ].
This“local” construction makes sense, as the following claim shows:
Claim 2. Let K be P-generic over V [H]. Then every real in V [H][K] is in a
model of the form M [H][K].
2κ
Proof of Claim 2. To any x ∈ ω ω ∩V [H][K] there is a name τ in V Col(ω,2 )×Col(ω,λ)
κ
such that τ H×K = x. Since 22 ≤ λ, this name can be chosen to be small, i.e.
τ ∈ Hλ+ . As κ is λ-strong, there exist M and π : V → M such, that τ ∈ M .
5
Hence x ∈ M [H][K].
(Claim 2)
For a fixed π we will construct a tree U ∗ in V [H]. We will then merge these
trees.
We fix an embedding π; there is a (κ, π(κ))-extender E derived from π. We set
νa := π(Ea ), then νa is a measure in M . For (s, a) ∈ ω n × κn the following holds:
(s, a) ∈ π(U ) ⇐⇒ a ∈ π(U )s = π(Us )
⇐⇒ Us ∩ κn ∈ Ea
⇐⇒ π(Us ∩ κn ) = π(Us ) ∩ π(κ)n ∈ νa .
In V [H] the set P(P({a ; a is a finite partial function ω → κ)}))V is countable,
so we can enumerate the sets νa in V [H].2 We choose an enumeration hσi | i ∈ ωi
in V [H], such that each νa appears infinitely often in the enumeration. We fix
a function g such that g : ω → ω; i 7→ dom(a) for an a such that νa = σi . The
function g is well defined, since for all a, b such that νa = σi = νb the measure
σi “concentrates” on one domain, i.e. dom(a) = dom(b) holds. Without loss of
generality we can assume g(i) ⊆ i for all i < ω. We say σk projects to σi if and
only if there is a ( b such that σi = νa and σk = νb with the property that
X ∈ νa ⇐⇒ X ab ∈ νb .
Hence for every i ∈ ω there is πi : M →σi Ult(M, σi ). If σk projects to σi , there
is an embedding πik : Ult(M, σi ) → Ult(M, σk ).
We can now (in V [H]) define the tree U ∗ on ω × π((2λ )+ ). We put
(s, (α0 , . . . , αn−1 )) ∈ U ∗
if and only if
∀i < k < n(π(Usg(i) ∩ κn ) ∈ σi ∧ π(Usg(k) ∩ κn ) ∈ σk ∧ σk projects to σi
→ πik (αi ) > αk ).
The tree U ∗ witnesses locally the universal Baireness of A, as the following claim
shows.
Claim 3. If K is P-generic over M [H], then
M [H][K] |= p[π(U )] = ω ω − p[U ∗ ].
Proof of Claim 3. We assume, there was x ∈ p[π(U )] ∩ p[U ∗ ] and work towards a
contradiction. Say (x, f ) ∈ [π(U )] and (x, α
~ ) ∈ [U ∗ ] where α
~ = hαi | i ∈ ωi. For
2
Note that the following can hold: dom(a) = dom(b) and a 6= b but νa = νb .
6
all n the following holds: π(Uxn ) ∩ π(κ)n ∈ νfn . If m < n it is evident that νfn
projects to νfm . We set
M̃ := dirlimn Ult(M ; νfn ).
We fix a sequence (in )n∈ω such that νfn = σin . By ρn we denote the canonical
embedding from Ult(M ; νfn ) to M̃ . Now
πin in+1 (αin ) > αin+1
⇐⇒ ρn+1 (πin in+1 (αin )) > ρn+1 (αin+1 )
⇐⇒ ρn (αin ) > ρn+1 (αin+1 ).
Hence the sequence hρn (αin ) | n ∈ ωi witnesses that M̃ is not well-founded. But
M̃ can embedded into Ult(M ; π(E)) by the universal property of the direct limit.
This is a contradiction to the wellfoundedness3 of Ult(M ; π(E)).
Let now x ∈
/ p[π(U )], so the tree π(U )x is well-founded. We have to show x ∈
∗
p[U ]. For n < ω and ~γ ∈ π(κ)n we define
fn (~γ ) = k~γ kπ(U )x
= the rank of ~γ in the tree order of π(U )x .
Every σi is a measure in M . We extend each σi to a measure σ̃i in M [H][K], by
setting
σ̃i := {X ; ∃X 0 ∈ σi X 0 ⊆ X}M [H][K] .
By this procedure we can extend the canonical embeddings πi : M → Ult(M ; σi )
to embeddings π̃i . Analogously we proceed with πik for any relevant i, k.
We can now put
αk = [fg(k) ]σ˜k
= the equivalence class of fg(k) in Ult(M [H][K]; σ˜k ).
Let α
~ = hαi | i ∈ ωi. We have to show (x, α
~ ) ∈ [U ∗ ]. Let π(Usg(i) ) ∈ σi ,
π(Usg(k) ) ∈ σk and let σk project to σi . Note that g(i) ( g(k). Then we have
πik (αi ) = π˜ik (αi )
= π˜ik ([~γ 7→ k~γ kπ(U )x ]σ˜i )
= [~ε 7→ k~ε g(i)kπ(U )x ]σ˜k
> [~ε 7→ k~εkπ(U )x ]σ˜k
= αk .
(Claim 3)
3
See remarks after [Jec03, 20.28].
7
We are now ready to merge the trees constructed so far. Let K be P-generic
over V [H]. By Claims 2 and 3 there is for every y = τ H×K ∈ V [H][K] ∩ ω ω an
∗
that witnesses in the following sense the
embedding π : V → M and a tree Uτ,π
λ-universal Baireness:
• y ∈ M [H][K]
∗
• M [H][K] |= p[π(U )] = ω ω − p[Uτ,π
].
∗
+
by (s, f + ) where
by replacing each (s, f ∗ ) ∈ Uτ,π
We now define disjoint trees Uτ,π
dom(f + ) = dom(f ∗ ) ∧ [∀n ∈ dom(f ∗ ) (f ∗ (n) = α ⇐⇒ f + (n) = (α, τ ))].
In V [H] we put
[
κ
+
U ∗ = {Uτ,π
; τ H is a Col(ω, 22 )×Col(ω, λ)-nice name for a real and τ ∈ Hλ+ }.
Clearly
p[U ∗ ] =
[
+
p[Uτ,π
]=
[
∗
p[Uτ,π
].
τ,π
τ,π
ω
Let K be P-generic over V [H] and x ∈ ω ∩ V [H][K]. We discuss the case
∗
], hence x is in none of the
x ∈ p[U ], then by Claim 3 x is in none of the p[Uτ,π
+
∗
p[Uτ,π ]. So x is not in p[U ] either.
∗
∗
Let now x ∈
/ p[U ]. By claims 2 and 3 there is Uτ,π
such that x ∈ p[Uτ,π
]. So
∗
x ∈ p[U ]. Hence
V [H]P |= p[U ] = ω ω − p[U ∗ ].
3
The 12th Delfino Problem
Lemma 2.3 allows us to analyse the consistency strength of the following statement: “Every projective set is Lebesgue measurable, has the Baire property and
has a projective uniformization”. More precisely we will give an (optimal) upper bound for the consistency strength. By a projective uniformization of a set
A ⊆ (ω ω )2 we understand a projective set F such that
F ⊆ A ∧ ∀x(∃y((x, y) ∈ A) ⇐⇒ ∃!y((x, y) ∈ F ))).
For the sake of brevity we define:
Definition 3.1 4 :≡ “Every projective set is Lebesgue measurable, has the
Baire property and every projective set in (ω ω )2 has a projective uniformization”.
8
Now we define the #12 hypothesis, a large cardinal axiom. The name #12
hypothesis stems from the 12th Delfino Problem. We will explicate the 12th
Delfino Problem below.
Definition 3.2 We say that the #12 hypothesis holds if and only if there are
cardinals κi , i ∈ ω, κi < κi+1 and lim κi = λ such that
∀i ∈ ω κi is λ-strong.
The #12 hypothesis is an upper bound for 4, i.e.:
Theorem 3.3 (Steel, 1997) If there is a model of “ZFC+ #12 hypothesis”, there
is a model of “ZFC+4”.
In the course of this chapter we will prove theorem 3.3. This theorem is closely
related to the 12th Delfino Problem. The 12th Delfino Problem was formulated
by Woodin in [Woo82] and is the following question:
Does 4 imply projective determinacy?
Another way of formulating this would be: If one considers 4 as the natural consequences4 of projective determinacy, do these natural consequences yield projective determinacy? Compare the list of Delfino Problems in [KMS88]. Steels
answer to this question is hence no, since the #12 hypothesis far weaker as projective determinacy, for projective determinacy implies (by a result of Woodin)
the existence of a model with one Woodin cardinal. But below one Woodin cardinal δ there are inaccessibly many δ-strong cardinals, so in particular countably
many. If the #12 hypothesis would yield a model of projective determinacy, we
had arrived at contradiction to Gödel’s incompleteness theorem. Later we will
prove more directly building on a result of Martin that projective determinacy is
not implied by the #12 hypothesis.
Proposition 3.4 Let κ0 < κ1 < · · · with limit λ be witnesses to the #12
hypothesis. Let G be Col(ω, λ)-generic over V . In V [G] every projective set is
Lebesgue measurable and has the Baire property.
Proof. We fix a sequence hGi | i ∈ ωi such that
κ
• G0 is Col(ω, 22 0 )-generic over V ,
• Gi+1 is Col(ω, 22
κi+1
)-generic over V [G0 , . . . , Gi ].
4
We see 4 as the natural consequences of projective determinacy since projective determinacy decides certain regularity properties of the projective sets, in particular 4 is a consequence
of projective determinacy. See [Kec95]
9
Note that all Gi can be chosen in V [G], since in V [G] there are only countably
κ
many dense subsets of Col(ω, 22 i ) in V [G0 , . . . , Gi−1 ], because λ is a strong limit.
Inductively we construct for i ≥ 1 trees Si , Ti with the following properties
• p[Ti ] is a universal Π1i set in every generic extension of size equal or less
than λ,
• Si , Ti ∈ V [G0 , . . . , Gi−2 ] for i ≥ 2, else Si , Ti ∈ V ,
• Si , Ti witness that A = p[Ti ] is λ-universally Baire in V [G0 , . . . , Gi−2 ] for
i ≥ 2 and in V respectively.
Let A ⊆ (ω ω )m be a universal Π11 set. By proposition 1.4 every coanalytic set is
universally Baire, where the γ-universal Baireness is witnessed by the Shoenfield
tree on ω m × 2γ for each γ. The Shoenfield tree T1 for A on ω m × 2λ can even be
chosen to be in L. By proposition 1.4 there is a “complementary” tree S1 .
Let A = p[Tn ] ⊆ (ω ω )m+2 be a universal Π1n set and n ≥ 1. We construct Tn+1
and Sn+1 now. We set
B̄ = {(~x, z) ; ∃y(~x, y, z) ∈ A}.
By lemma 2.3 there are trees Tn+1 and Sn+1 such that B̄ = p[Sn+1 ] and (ω ω )m+1 −
p[Sn+1 ] = p[Tn+1 ] in every size ≤ λ generic extension of V [G0 , . . . , Gn−1 ]. We have
to check that p[Tn+1 ] is a universal Π1n+1 set in every generic extension of size ≤ λ.
We first discuss an easier case: let C̄ ⊆ (ω ω )m be a Π1n+1 set in V [G0 , . . . , Gn−1 ].
Then
(ω ω )m − C̄ = ∃R C
for a Π1n set C. Then there is a real z such that
(ω ω )m − C̄ = {~x ;
= {~x ;
= {~x ;
= {~x ;
∃y(~x, y) ∈ C}
∃y(~x, y, z) ∈ A}
(~x, z) ∈ B̄]}
(~x, z) ∈
/ p[Tn+1 ]}.
This implies
C̄ = {~x ; (~x, z) ∈ p[Tn+1 ]}.
If C is a set in a generic extension of V [G0 , . . . , Gn−1 ] of size ≤ λ we find the
real parameter z by our inductive hypothesis (the point is, that z is a real in
the generic extension). By the product lemma the forcing extension V [G] is (in
particular) a forcing extension of V [G0 , . . . , Gi ] of size ≤ λ for every i ∈ ω. If
H is a further generic object to a forcing of size ≤ λ in V [G], then V [G][H] is a
forcing extension of V [G0 , . . . , Gi ] of size ≤ λ for every i ∈ ω too. The universal
Π1n sets constructed above are universal in V [G]. In particular we can construct
10
trees out of the trees for these universal sets for every projective set: If A ⊆ ω ω
is a Π1n set, say A = {x ; (x, y) ∈ p[Tn ]} for a fixed y ∈ ω ω , we set
T = {(s, f ) ∈ ω <ω × (2λ )<ω ; ∃t ⊂ y(s, t, f ) ∈ Tn }.
Let S be analogously defined. Clearly p[T ] = ω ω − p[S] in every generic extension
of size ≤ λ and p[T ] = A.
Hence every projective set in V [G] is λ-universally Baire, so in particular by
proposition 1.5 it has the property of Baire.
The λ-universal Baireness of every projective set in V [G] is too weak, to infer the
Lebesgue measurability directly from proposition 1.5. We thus resort to methods
of Solovay. We remind the reader of the following (compare [Jec03, 26.4, 26.6]):
Definition and Remark 3.5 A set A ⊂ ω ω is Solovay over a transitive model
N of set theory, if there is a formula ϕ and there are parameters ~s ∈ N such that
x ∈ A ⇐⇒ N [x] |= ϕ(x, ~s).
If A, N , ~s are all as above, then we have:
S
1. If the set {B ; B is a Borel null set with code in N } is null, the set A is
Lebesgue measurable.
2. In particular: If N ∩ ω ω is countable, then A is Lebesgue measurable, since
there are only countably many Borel codes for null sets.
Starting with a projective set we will construct a transitive N such that V [G]
believes that N ∩ ω ω is countable. Let A be a Π1n set, say A ∈ Π1n ({y}) for
y ∈ V [G] ∩ ω ω ; without loss of generality we assume:
x ∈ A ⇐⇒ (x, y) ∈ p[Tn ].
Let ẏ ∈ V be a name for y. Without loss of generality we assume ẏ ∈ Hλ+ .
The ordinal κn−1 is λ-strong in V [G0 , . . . , Gn−2 ]. So there is an extender E with
critical point κn−1 and
π : V [G0 , . . . , Gn−2 ] →E M [G0 , . . . , Gn−2 ] =: M̃
such that ẏ ∈ M̃ . Without loss of generality assume that M is the ultrapower
of V by E, we can arrange this in this fashion by [Jec03, 20.30]. Since y is in
a generic extension of M̃ , there is, by [Jec03, 15.42], a forcing Q such that y is
Q-generic over M̃ . So the model M̃ [y] is well defined.
Claim 1. The reals of N := M̃ [y] are countable in V [G].
Proof of Claim 1. We will proof this for the case n = 1. So
π : V →E Ult(V, E) = M̃
11
for an extender E with crit(E) = κ := κ0 . By definition π(κ) > λ. We now show
π(κ) < λ+ . The elements of M̃ can be represented by equivalence classes of the
form [a, f ] for a ∈ λ<ω and f : κ|a| → V . By the properties of these equivalence
classes the following holds:
π(κ) = {[a, f ] ; [a, f ] ∈ π(κ)}
= {[a, f ] ; f [X] ⊆ κ for some X ∈ Ea }
= {[a, f ] ; f : κ|a| → κ}.
Now we can calculate the cardinality (in V ) of π(κ):
λ ≤ Card({[a, f ] ; f : κ|a| → κ}) ≤ Card(λ<ω ) · κκ = λ · 2κ = λ,
where the last equality holds since λ is a strong limit. Then we have
M̃ |= λ < π(κ) < λ+ ∧ π(κ) is a strong limit,
and hence
M̃ |= Card(Col(ω, λ)) < π(κ).
Thus it follows that there are less than π(κ) many nice names for reals for
Col(ω, λ) in M̃ , in particular less than π(κ) many nice names for reals for Q.
Then there are less than π(κ) many reals in N = M̃ [y]. Note that the proof
V [G]
works in V [G] as well. We have π(κ) < λ+ = ω1 , hence the reals of N are
countable in V [G].
If n > 1 we can use the same proof as above, since collapsing finitely many
κi < κn−1 does not change the relevant properties of the ultrapowers involved.
(Claim 1)
It remains to show that A is Solovay over N .
Claim 2. x ∈ A ⇐⇒ M̃ [y][x] |= (x, y) ∈ p[π(Tn )].
Proof of Claim 2. Note that models of the form M̃ [y][x] are well defined by [Jec03,
15.42] and the fact, that there is always a smallest transitive model containing
M̃ [y] and x. The proof of this claim is nearly the same as the proof of claim 1 in
the proof of lemma 2.3. Let x ∈ A, say
∀n(x n, y n, f n) ∈ Tn .
Then the following holds too:
∀n(x n, y n, π(f n)) ∈ π(Tn ).
So (x, y) ∈ p[π(Tn )].
Let now x ∈
/ A. Then (x, y) ∈ p[Sn ]. By an analogous argument as above
(x, y) ∈ p[π(Sn )]. Since
V [G0 , . . . , Gn−2 ] |= p[Tn ] is λ-universally Baire, witnessed by Tn and Sn .
12
implies
M̃ |= p[π(Tn )] is π(λ)-universally Baire, witnessed by π(Tn ) and π(Sn ),
in particular
M̃ |= p[π(Tn )] ∩ p[π(Sn )] = ∅ in all generic extensions of size ≤ λ.
The absoluteness of wellfoundedness implies:
V [G] |= p[π(Tn )] ∩ p[π(Sn )] = ∅.
So
M̃ [y][x] |= (x, y) ∈
/ p[π(Tn )].
(Claim 2)
This finishes the proof of the proposition.
In the remainder of this chapter we will prove theorem 3.3 and we will explain,
why this theorem solves the 12th Delfino Problem. It is an open problem whether
all projective sets in the model V [G] just discussed have a projective uniformization. To proof theorem 3.3, we must replace V by an adequate core model. We
use the minimal, fully iterable, fine structural inner model L[E], that satisifes the
#12 hypothesis. The construction of such a model would go beyond the scope
of this paper. We can not give proof of all properties of this model we need. For
a construction of L[E]-type models see [Sch02], [Steb], [Stea] and [MS94]. From
now on until the end of this paper let V = L[E]. As in the preceding proposition
we fix a filter G that is Col(ω, λ)-generic over L[E].
An important property of L[E] is that it is the core model (in a precise sense) of
all its set sized generic extensions. Therefore we write L[E] = K too. We will
not further elaborate this property of K, since we will not work out any details
of proofs that rely on this fact.
Remark 3.6 In K[G] there is a non determined projective set, i.e. ¬(∆12
determinacy) holds in K[G], as we will see later. This is why theorem 3.3 is a
solution to the 12th Delfino Problem.
So we must show that in L[E][G] = V [G] all projective sets in (ω ω )2 have
a projective uniformization. We fix a projective set A ⊂ (ω ω )2 , A ∈ Π1n , say
A ∈ Π1n ({z0 }) for a z0 ∈ ω ω . Let Tn , Sn on ω 3 × 2λ be as in the proof of
proposition 3.4, then Tn , Sn ∈ V [G0 , . . . Gn−2 ]5 . Without loss of generality let
A = {(x, y) ; (x, y, z0 ) ∈ p[Tn ]}.
5
By theorem 1.6 we could assume T2 , S2 ∈ V and Tn , Sn ∈ V [G0 , . . . , Gn−3 ] for larger
n. This is a nice fact when calculating the complexity of the uniformization. But since we
have neither proved theorem 1.6 nor will we calculate the complexity of the uniformization,
T2 , S 2 ∈
/ V will not do any harm.
13
Lemma 3.7 The construction of Tn , Sn can be modified to yield: In every generic
extension of V [G0 , . . . Gn−2 ] of size < κn−1 the following holds
• p[Tn κn−1 ] is the universal Π1n set in (ω ω )3 ,
• p[Tn κn−1 ] = (ω ω )3 − p[Sn κn−1 ],
where Tn κn−1 := {(s, t, u, f ) ∈ Tn ; ran(f ) ⊂ κn−1 } and Sn κn−1 := {(s, t, u, f ) ∈
Sn ; ran(f ) ⊂ κn−1 }
Proof. The point is to change the basis of the induction. Since κn−1 is in particular a strong limit, the Shoenfield tree of a Π11 set on (ω)3 × κn−1 witnesses the
γ-universal Baireness for every γ < κn−1 . As in the proof of propostion 3.4 we
have at the basis of the induction
p[T1 κn−1 ] = (ω ω )3 − p[S1 κn−1 ].
If we choose a universal Π11 set p[T1 κn−1 ] at the basis of the induction, a
construction as in the proof of 3.4 yields the desired conclusion.
We fix x ∈ ω ω . We will find, in a projective and uniform way, a real F (x)
such that
∃y(x, y) ∈ A =⇒ ∃y((x, y) ∈ A ∧ y = F (x)),
So
∃y(x, y, z0 ) ∈ p[Tn ] =⇒ (x, F (x), z0 ) ∈ p[Tn ].
We find F by analysing certain premice. These premice correspond to reals by
coding. Thus we will be able to identify projective sets with sets of these mice.
This will yield a projective uniformization in the end. To define the relevant class
of premice we need the following notion:
Definition 3.8 A pair ((M0 , M1 ), κ) is a phalanx, if and only if M0 , M1 are
premice and the following conditions are met:
• κ ∈ M0 ∩ M1 is a cardinal in M1 ;
• E M0 κ = E M1 κ;
• if κ0 < κ, then κ0 is a cardinal in M0 if and only if κ0 is a cardinal in M1 .
We call κ the exchange point of the phalanx. If it is clear which exchange point
is intended we write, for the sake of brevity, (M0 , M1 ) instead of ((M0 , M1 ), κ).
There is a notion of the iterability of a phalanx and of coiterations of two phalanges. Since these are the kind of details we are not really concerned here with,
we refer to [Zem02, 9.1] and [Ste96] for these notions.
14
Remark 3.9 In our situation iterations of phalanges are not anymore linear,
since the iterations of premice are not either. Hence we need iteration trees. If
we use a Jensen style indexing, these trees are almost linear: see [Sch02]. We
suppress all details of these iteration trees in the context of this paper.
Definition 3.10 A premice M is called (z0 , x)-good, if all of the following conditions are met:
1. M . JκK+K .
n−1
2. The phalanx ((K, M ), κ+K
n−1 ) is iterable.
3. M has a topextender F M with critical point κn−1 .
4. If π : M →F M M̃ and π̃ : M [G0 , . . . , Gn−2 ] → M̃ [G0 , . . . , Gn−2 ] is the canonical extension of π, then
∃y(x, y, z0 ) ∈ p[π̃(Tn κn−1 )].
K
Note that Tn κ+K
n−1 ∈ Jκ+K ∈ M .
n−1
5. M is minimal (in respect to the canonical prewellordering of premice) among
all mice satisfying 1. through 4.
Remark 3.11 In V [G] there are (z0 , x)-good premice which are countable, as
we will see below. The fifth condition makes sense, since the second condition
implies the iterability of the premouse. We will not prove this fact here.
Definition 3.12 Let M0 be a premice and T an iteration tree on M0 . If 0T α,
we say that Mα is an iterate of M0 . If
[0, α]T ∩ DT = ∅,
i.e. there are no drops on the branch through T from 0 to α, then we say
that Mα is a simple iterate of M0 . Note that there is an elementary embedding
i0,α : M0 → Mα in this case.
We will use the following lemma without proof, since such a proof would go
beyond the scope of this paper.
Lemma 3.13 If M , M 0 are (z0 , x)-good premice, then M , M 0 coiterate simply
above κ+K
n−1 .
We examine now, which iterates of a (z0 , x)-good premice are (z0 , x)-good.
15
Lemma 3.14 Simple iterates above κ+K
n−1 of a (z0 , x)-good premouse are (z0 , x)good.
Proof. Let M be (z0 , x)-good and M ∗ a simple iterate of M above κ+K
n−1 . Let
i : M → M ∗ denote the embedding. We verify the definition step by step.
∗
K
1. Since the iteration is above κ+K
n−1 , clearly M . Jκ+K holds.
n−1
2. The phalanx (K, M ∗ ) is an iterate of the phalanx (K, M ), so it is iterable,
since any iteration of (K, M ∗ ) is an iteration of (K, M ).
3. Clearly i(κn−1 ) = κn−1 . It may be the case that ht(M ) and ht(M ∗ ) differ,
∗
but the image of M ’s topextender F M has critical point κn−1 too.
4. In an abuse of notation we write π for ultrapower map of the ultrapower of
∗
M ∗ by F M too. As in the case for M , the map π induces a map
π̃ : M ∗ [G0 , . . . , Gn−2 ] → M̃ ∗ [G0 , . . . , Gn−2 ].
∗
Since F M is the image of M ’s topextender, we get another map
ĩ : M [G0 , . . . , Gn−2 ] → M ∗ [G0 , . . . , Gn−2 ].
By the shift lemma (see [Ste96]) there is a map j and the following commutative diagram:
- M̃
M
π
i
j
?
M
∗
π
?
- M̃ ∗
Then there is a commutative diagram with the extended maps ĩ, j̃ and π̃:
M [G0 , . . . , Gn−2 ]
- M̃ [G0 , . . . , Gn−2 ]
π̃
ĩ
j̃
?
∗
M̃ [G0 , . . . , Gn−2 ].
M [G0 , . . . , Gn−2 ]
∗
?
π̃
Then clearly the following holds on the M ∗ -side:
∃y(x, y, z0 ) ∈ p[π̃(Tn κn−1 )].
16
5. Since M ∗ is a simple iterate of M , both are in the same equivalence class
in respect to the canonical prewellordering of mice.
The following lemma will yield in the end that the uniformization F we are
looking for is projective.
Lemma 3.15 The set of all reals that code (z0 , x)-good premice is projective.
Proof. We verify that we can formulate the definition in a projective way step by
step.
1. JκK+K is countable in V [G], so we can code it into a real. We can code
n−1
structures of the form M ||δ by reals too.
2. The iterability of a phalanx (K, M ) is a projective statement in the codes
by results of Hauser, see [Hau95, 3.4].
3. Clearly the following statement is projective in the codes: “M is a countable
premouse with a topextender with critical point κn−1 ”. Note that κn−1 is
countable in V [G].
4. The filters G0 , . . . , Gn−2 are countable in V [G]. So after coding we can
identify them with reals. The ultrapower of a countable (z0 , x)-good M by
its topextender is countable. Since we can use the codes of G0 , . . . , Gn−2
as real parameters, we can code the map π̃ in definition 3.10 by a real. In
particular we can code π̃(Tn κn−1 ) by a real. Thus
∃y(x, y, z0 ) ∈ p[π̃(Tn κn−1 )].
is projective in the codes.
5. Coiterations of countable premice are basically countable sequences of countable well founded models, so we can code coiterations by reals. Hence we
can formulate the minimality in respect to the canonical prewellordering of
premice of some M in a projective way in the codes. Thus we can state
projectively that M is minimal among all premice satisfying conditions 1.
through 4.
The following lemma allows us to analyse membership in A by looking at
(z0 , x)-good premice.
17
Lemma 3.16
∃y(x, y, z0 ) ∈ p[Tn ] ⇐⇒ ∃y(x, y) ∈ A
⇐⇒ There is a countable (z0 , x)-good M.
Proof. The first equivalence clearly holds by our choice of z0 . We proceed as in
the proof of lemma 2.3: Let y ∈ ω ω such that (x, y, z0 ) ∈ p[Tn ]. Then there is an
initial segment M of K with a topextender F M with critical point κn−1 such that
the tuple (x, y, z0 ) is in a generic extension of size < π̃(κn−1 ) of M̃ [G0 , . . . Gn−2 ].
The initial segment M is iterable by basic properties of K. Without loss of
generality we can choose M to be countable in V [G]. The phalanx (K, M ) is
iterable. We have to show
(x, y, z0 ) ∈ p[π̃(Tn κn−1 )],
then M witnesses the existence of a (z0 , x)-good premouse. We assume
(x, y, z0 ) ∈
/ p[π̃(Tn κn−1 )]
and work towards a contradiction. Since Tn κn−1 and Sn κn−1 are trees to a
γ-universally Baire set, for all γ < κn−1 , we have
(x, y, z0 ) ∈ p[π̃(Sn κn−1 )].
We can form the ultrapower of V [G0 , . . . , Gn−2 ] by F M . For the ultrapower map
we write in an abuse of notation π̃. Then
(x, y, z0 ) ∈ p[π̃(Sn κn−1 )] ⊆ p[π̃(Sn )].
Since (x, y, z0 ) ∈ p[Tn ] and reals are not moved by π̃ we have (x, y, z0 ) ∈ p[π̃(Tn )].
But then
p[π̃(Tn )] ∩ p[π̃(Sn )] 6= ∅,
which implies
p[Tn ] ∩ p[Sn ] 6= ∅.
Contradiction!
Let M be a (z0 , x)-good premouse with topextender F M . As in the proof of
the converse direction we extend π̃ to a map with domain V [G0 , . . . , Gn−2 ]. Let
y ∈ ω ω such that
(x, y, z0 ) ∈ p[π̃(Tn κn−1 )] ⊂ p[π̃(Tn )].
If (x, y, z0 ) ∈
/ p[Tn ] would hold true, then (x, y, z0 ) ∈ p[Sn ] and so (x, y, z0 ) ∈
p[π̃(Sn )]. So this would yield the same contradiction as above!
18
For any given (z0 , x)-good M and any y, α
~ ∈ M such that
(x, y, z0 , α
~ ) ∈ [π̃(Tn κn−1 )]
there is a leftmost branch (y, α
~ ). This branch is not necessarily a honest leftmost
branch in the sense of [Kan03].
Definition 3.17 We write (y, α
~ )M for the leftmost branch in M witnessing
∃y(x, y, z0 ) ∈ p[π̃(Tn κn−1 )].
Definition 3.18 We define for any (z0 , x)-good N
B(N ) := {N ∗ ; N ∗ is a simple iterate above κ+K
n−1 of N }.
By lemma 3.14 the set B(N ) contains only (z0 , x)-good premice. A (z0 , x)-good
M will be called stable if and only if for all M ∗ ∈ B(M ) with iteration map
i : M → M ∗ the following holds
y ≤lex y ∗ ∧ (∀n ∈ ω)i(α(n)) ≤ α∗ (n),
∗
where (y, α
~ ) = (y, α
~ )M and (y ∗ , α
~ ∗ ) = (y, α
~ )M .
Lemma 3.19 If there is a (z0 , x)-good M , then there is a stable (z0 , x)-good M .
If M and M 0 are stable (z0 , x)-good premice, then y = y 0 , where (y, α
~ ) = (y, α
~ )M
0
and (y 0 , α
~ 0 ) = (y, α
~ )M .
This lemma finishes the proof of theorem 3.3; we can now define the uniformization F of A by setting
(x, y) ∈ F ⇐⇒ x ∈ A ∧ ∃M ∃~
α(M is stable (z0 , x)-good ∧ (y, α
~ ) = (y, α
~ )M ).
“M is a (z0 , x)-good premouse” is a projective statement in the codes. Note that
we mentioned above, that statements about countable iterations of countable
premice are projective in the codes. By the preceding remarks and lemmata F is
a projective set. So we have found a projective uniformization of A. It remains
to proof lemma 3.19.
Proof. Let M be (z0 , x)-good. We choose M1 ∈ B(M ) such that for all M1∗ ∈
∗
B(M1 ) the following holds true: If (y, α
~ ) = (y, α
~ )M1 and (y ∗ , α
~ ∗ ) = (y, α
~ )M1 , then
y(0) ≤ y ∗ (0). Here y(0) is minimal among all (z0 , x)-good N , since if Q denotes
the result of a coiteration of M with a (z0 , x)-good N , then Q ∈ B(M ) by lemma
3.13.
We hone this procedure to minimise α
~ (0). Let (y1 , α
~ 1 ) = (y, α
~ )M1 . We choose, if
it exists, M1,1 ∈ B(M1 ) such that
α1,1 (0) < iM1 ,M1,1 (α1 (0)),
19
where iM1 ,M1,1 denotes the iteration map and (y1,1 , α
~ 1,1 ) = (y, α
~ )M1,1 . We proceed inductively to construct, if it exists, a (maybe finite) sequence of models
M1,n+1 such that M1,n+1 ∈ B(M1,n ) and α1,n+1 (0) < iM1,n ,M1,n+1 (α1,n (0)), where
(y1,n , α
~ 1,n ) = (y, α
~ )M1,n and iM1,n ,M1,n+1 is the according iteration map.
Claim 1. The sequence of the M1,n is finite.
Proof of Claim 1. We assume that this was not the case. Then the direct limit
of the M1,n is an iterate of M1 . But since α1,n+1 (0) < iM1,n ,M1,n+1 (α1,n (0)) holds
true for all n, this direct limit is not well founded, contradicting the iterability of
(Claim 1)
M1 .
We set M2 to be the last model in the sequence of the M1,n or M1 itself if the
sequence is empty. This construction obviously yields for all N ∈ B(M2 )
i(α2 (0)) ≤ αN (0),
where i : M2 → N denotes the iteration map, (y2 , α
~ 2 ) = (y, α
~ )M2 and (yN , α
~N) =
N
(y, α
~ ) . We repeat these to steps to produce a sequence (Mi )i∈ω : If i = 2j +1 , we
minimize y(j); if i = 2j, we minimize α
~ (j). In each case we use the procedures
described above for M1 and M2 respectively. Since for every n ≤ m ∈ ω the
model Mm is a simple iterate of Mn , the set of the Mi is a directed system. Let
Mω denote the direct limit of the Mi . Then Mω is a simple iterate above κ+K
n−1 of
M , so it is (z0 , x)-good. Clearly Mω is stable.
This lemma completes the proof of theorem 3.3. We will now look again at
the model K[G] and analyse why it is not a model of projective determinacy. We
will see that there is a ∆12 set in K[G] that is not determined.
We will say a few words about games and strategies. We see strategies in games
of the type G(ω, A) as (partial) functions σ : ω <ω → ω. See [Kec95, 20.A]. These
strategies can obviously be coded by reals. If σ is a real, that codes a strategy
for II and if x ∈ ω ω is the real that I plays, then σ ∗ x denotes the real II plays
by using the strategy σ on x. If τ codes a strategy for I, the we write x ∗ τ for
the real that I plays by using τ on an x played by II. Compare [Kec95, 39.1].
Theorem 3.20 There is a non determined ∆12 set in K[G].
We noted above that by an unpublished result of Woodin we already know
consistency strength wise that ∆12 determinacy can not hold in K[G], but the
proof below is more elementary and gives some insight into the nature of that
non determined set. The proof shows that a non determined ∆12 set in K induces
(by the same definition) a non determined ∆12 set in K[G]. First of all we need
a general proposition on determinacy and regularity properties of sets of reals.
20
Proposition 3.21 If ∆12n determinacy holds, then all Σ12n+1 sets are Lebesgue
measurable, have the Baire property and have a perfect subset or are countable.
Proof. By a result of Martin ∆12n determinacy implies Π12n determinacy. For a
proof of this result see [KS85, 5.1]. By a result of Kechris and Martin (about
unfolded games) this implies already the desired conclusion. For a proof of this
result see [Kan03, 27.14].
We proof theorem 3.20 now.
Proof. By lemma 2.3 every projective set of reals in K[G] is λ-universally Baire.
In particular all ∆12 sets are λ-universally Baire. By theorem 1.7 we know
K ≺Σ13 K[G].
In K there is a Σ13 wellordering of the reals. Clearly this wellordering is not
measurable. So by proposition 3.21 there is a non determined ∆12 set A in K.
Let ϕ and ψ be Σ12 formulas and y ∈ ω ω ∩ K a parameter with
K |= x ∈ A ⇐⇒ ϕ(x, y) ⇐⇒ ¬ψ(x, y).
The statement
∀x(ϕ(x, y) ⇐⇒ ¬ψ(x, y))
is Π13 . So
K[G] |= ∀x(ϕ(x, y) ⇐⇒ ¬ψ(x, y)).
Claim 1. The non determinacy of A is a Π13 statement.
Proof of Claim 1. Since A is not determined for every strategy τ of player I there
is an x ∈ ω ω such that τ ∗ x ∈
/ A and analogously for II. The following statement
1
is Π3 and expresses the non determinacy of A in K:
(∀τ ∃x1 ψ(x1 ∗ τ, y)) ∧ (∀σ∃x2 ϕ(σ ∗ x2 , y)).
(Claim 1)
So the following holds true in K[G]
(∀τ ∃x1 ψ(x1 ∗ τ, y)) ∧ (∀σ∃x2 ϕ(σ ∗ x2 , y)) ∧ ∀x(ϕ(x, y) ⇐⇒ ¬ψ(x, y)).
The set Ā = {x ∈ ω ω ∩ K[G] ; ϕ(x, y)} is then complementary to {x ∈ ω ω ∩
K[G] ; ψ(x, y)} in K[G] and not determined.
To sum it up:
K[G] |= 4 + ¬(∆12 determinacy).
This completes our examination of the 12th Delfino Problem. We remark that
our upper bound for 4 is the best possible.
21
Remark 3.22 With core model theory one can show that the #12 hypothesis
and 4 are equiconsistent, see [Sch02, 9.1].
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23